Limits of Defect Tolerance in Perovskite Nanocrystals: Effect of Local Electrostatic Potential on Trap States

One of the most promising properties of lead halide perovskite nanocrystals (NCs) is their defect tolerance. It is often argued that, due to the electronic structure of the conduction and valence bands, undercoordinated ions can only form localized levels inside or close to the band edges (i.e., shallow traps). However, multiple studies have shown that dangling bonds on surface Br– can still create deep trap states. Here, we argue that the traditional picture of defect tolerance is incomplete and that deep Br– traps can be explained by considering the local environment of the trap states. Using density functional theory calculations, we show that surface Br– sites experience a destabilizing local electrostatic potential that pushes their dangling orbitals into the bandgap. These deep trap states can be electrostatically passivated through the addition of ions that stabilize the dangling orbitals via ionic interactions without covalently binding to the NC surface. These results shed light on the formation of deep traps in perovskite NCs and provide strategies to remove them from the bandgap.

Computational methods. Geometry optimizations were carried out at the DFT level with a PBE exchange-correlation functional 1 and double-ζ basis set, as implemented in the CP2K quantum chemistry software package. 2 Relativistic effects were taken into account using effective core potentials. Due to the large size of the systems, spin-orbit coupling was not included.
The inverse participation ratio (IPR) and crystal orbital overlap population (COOP) were computed for the 500 highest occupied and 500 lowest unoccupied orbitals of each model. The IPR was loops over the AOs of all Br. A negative COOP value indicates that an MO is of an overall antibonding character, while positive COOP values indicate an overall bonding MO. COOP values close to zero are indicative of a non-bonding MO. Both the IPR and the COOP were calculated using the workflows implemented in the Nano-QMFlows package. 3 The external potential, mentioned in the main text during the discussion of Figure 3, was applied using the "EXTERNAL_POTENTIAL" keyword in CP2K. We introduced a Gaussian potential, centered 3.3 Å away from Br 1 (see Figure 3 in the main text), and described by . Here, is the maximal amplitude of the external potential (which we varied between -0.1 and 0.1 Ha in Figure S6) and , , are the spatial coordinates in Å. This potential is added as an extra term to the Hamiltonian. As a result, a positive external potential (i.e., ) lowers the energy of the electrons. > 0 The total potential energy experienced per MO (as plotted in Figures S3C, S4C   Ligand stripping. Following the method by Bodnarchuk et al., 4 we remove the outer CsBr layer of the model system M1 (shown in Figure S1) in four consecutive steps, in which we remove 25%, 50%, 75% and 100% of the CsBr units. This is shown in models M2-M5 in the first half of Figure S2. After removal of all CsBr units, the NC is terminated by a layer of PbBr 2 , with some loose Brleft on the NC surface. These loose Brare removed in model M6. Subsequently, in four consecutive steps, we again remove 25%, 50%, 75% and 100% of the PbBr 2 shell (models M7-M10, shown in the second half of Figure S2 on the next page).  Shallow traps in model M7. As discussed in the main text, removal of 25% of the PbBr 2 layer leads to the creation of Br --localized levels at the VB edge, which are shown in more detail in Figure S3. The highest five occupied levels are located on the PbBr 2 unit indicated in turquoise in Figure S3A. In the HOMO, there is a slight anti-bonding interaction between the Brand Pb 2+ , as indicated by the small Pb-contribution in the DOS and the negative COOP-value in Figure S3C. The four levels below the HOMO (i.e., HOMO-1 through HOMO-4) do not interact with Pb 2+ , as evinced by the COOP values of ~0. However, as can be seen in Figure S3B, the two Brions do interact with each other, forming π and π* bonds. As the π-and π*-like orbitals lie very close in energy, the Br --Brinteraction is expected to be small. Since the trap levels are close to the VB edge, these results are still in agreement with the concept of defect tolerance as discussed in the main text. However, further stripping of the PbBr 2 layer creates Br --localized levels that do lie in the middle of the bandgap (see Figure S4 and the main text).  Effect of removing PbBr 2 in different configurations. As discussed in the main text, the occurrence of deep Brtraps in model M8 is not linked to the specific structure of M8. Instead, it can be generalized to many different surface configurations, as long as Brions with only Cs + neighbors are present. To show Brtraps are not unique to model M8, we demonstrate in Figure S5 that removing 50% of the PbBr 2 moieties from different surface sites can also lead to the formation of deep traps.
Model M8, which is used in the main text (and also plotted in Figure S2), is shown in the leftmost column. Following the approach of Bodnarchuk et al., 4 PbBr 2 moieties were removed first from the corners and edges of the NC. In model M8-ii, PbBr 2 was first removed from the center of each facet. As shown in Figure S5A, this surface configuration is less favorable, increasing the total energy by 1.0 eV (compared to model M8). Moreover, all Brare coordinated to Pb 2+ , so that no traps are present in Figure S5D, confirming our interpretation of the traps as arising from Brwith no coordination to Pb 2+ . Moving one Brfrom model M8-ii away from its Pb 2+ neighbors results in structure M8-iii (the red circle indicates which Brhas been moved). In line with the results shown in the main text, this Br -, which now only has Cs + neighbors, experiences a significantly higher potential energy (see Figure  S5C). As also described in the main text, this leads to the formation of three deep traps, which are indicated in turquoise in Figure S5D. Yet, note that displacing the Brincreases the energy of the system by 0.4 eV, which makes the formation of this specific trap energetically unfavorable. In model M8-iv, surface PbBr 2 moieties have been removed from random positions. No traps are found due to the coordination of all surface Brto Pb 2+ . Again, moving one Br -(indicated by the orange circle) away from Pb 2+ causes the Brto experience a higher local potential energy, leading to the formation of three traps. However, in contrast to model M8-iii, there is only a negligible change in the total energy, indicating that both systems are roughly equally likely to be sampled at room temperature.
These results show that, although model M8 as used in the main text is not the lowest energy configuration, the formation of deep Brtraps can be generalized to many different PbBr 2 configurations, as long as Brions with only Cs + neighbors are present. Changes in the surface configuration can also lead to changes in the total energy, but the creation of a Brtrap does not necessarily lead to a significant increase of the energy of the system. Hence, thermal motion will cause the NC to sample a great number of surface compositions, with varying numbers of Brtraps. Shifting energy of traps. As discussed in the main text and shown in Figure 3B, application of a stabilizing external potential around Br 1 can push the energy of the traps back into the VB. Figure S6 shows that by varying the magnitude of the applied potential, the traps localized on Br 1 can be shifted through the entire bandgap. Figure S6A plots the energy (relative to the VB edge) of the three traps localized on Br 1 (shown in black, red, and blue) against the total potential energy experienced by those traps (see Computational methods for details). It can be seen that a more negative total potential energy lowers the energy of the trap with respect to the VB, and eventually pushes the trap below the VB edge (as indicated by the negative energy in Figure S6A). Conversely, a less negative total potential energy increases the energy of the traps with respect to the VB, and pushes them towards the CB, as also illustrated in Figure S6B.
The fits in Figure S6A show a strong linear correlation between the energy of the traps vs the VB edge and their total potential energy, with a slope in the range of 0.8-1.0. This suggests that the shift in energy is largely caused by the change in total potential energy, but that small additional effects take place in parallel, leading to a slope of not exactly 1. One significant outlier, in red at a total potential energy of -17 eV, may be explained by the fact that this trap, which lies in the VB, is slightly mixed with other VB levels. This is expected to lead to additional energy effects that break up the linear correlation. Electrostatic trap passivation. In order to investigate if deep traps can be removed solely through the electrostatic effect of ligands (i.e., without covalent binding to, and splitting of the non-bonding Br --orbital), we carried out two types of calculations. In the first, we constructed a sort of core/shell structure, based on model M1, as shown in Figure S7A. Here, the shell (indicated in gray) consists of all the atoms of model M1 that are removed during the stripping procedure ( Figure S2) to create model M8. The core (indicated with the standard colors for Cs, Pb and Br) comprises the same atoms that are in model M8. However, note that the coordinates of the core atoms in Figure S7A are not exactly the same as in model M8, since the coordinates of the former are extracted from the geometry optimization of model M1 in Figure S2. As a result, the atoms at the surface of the core (in Figure  S7A) still have the same configuration as they had in the bulk of model M1. Carrying out a single point calculation on the core atoms only, as shown in Figure S7D "without shell", indicates that this type of NC termination leads to many trap states and virtually no bandgap.
In the main text, we argued that the different potential experienced at the surface, causes Brdangling bonds to end up in the bandgap. By extension, one would expect that the traps in Figure S7D will be pushed out of the bandgap if the potential experienced at the core surface is equal to the potential experienced in the bulk. To simulate this effect, we removed the basis functions and electrons of all the atoms in the shell (gray in Figure S7A), leaving only the effective core potentials (see Computational methods). We corrected the charge of the core potentials so that the atoms in the shell have a charge of Cs + , Pb 2+ and Br -. As a result, the atoms in the shell mimic the perovskite bulk potential at the surface of the core, without forming any covalent bonds with the core surface. Figure  S7D shows that addition of this "point charge" shell effectively removes all traps from the bandgap. Figures S7B and S7C show that the band edges are delocalized over the entire core. Note that the MOs do not delocalize over the shell, as these atoms do not have basis functions. In the second calculation, the traps related to Br 1 were pushed out of the bandgap by placing a proton (H + )-like charge in the vicinity of Br 1 (see Figure S8A). To ensure no covalent binding to Br 1 takes place, we removed the basis functions of the proton. In addition, we removed the electron of the H + by setting the charge of the overall system (i.e., NC + proton) to +1, leaving only the effective core potential of the proton (with a total charge of +1). Figure S8B shows that the addition of the H + lowers the potential energy around Br 1 , which pushes the trap states on Br 1 back into the VB, as shown in red in Figure S8C. As the H + is only placed in the vicinity of Br 1 , the other four Brsites (indicated in turquoise in Figure S8A) still experience a high potential energy, which leaves their traps in the bandgap.